nLab exponentiable topos

Idea

An exponentiable topos is a generalization of the notion of an exponentiable locale and can be viewed as a (topological) “space” $X$ that behaves well with respect to the construction of mapping spaces $Y^X$.

Definition

Definition

A Grothendieck topos $\mathcal{E}$ is called exponentiable (in the 2-category of Grothendieck toposes $GrTop$) if the 2-functor ${}_{-}\times\mathcal{E}$ has a right 2-adjoint $(_{-})^\mathcal{E}$.

Remarks

• More concretely, $\mathcal{E}$ is exponentiable if there exists a functor $(_{-})^\mathcal{E}$ such that for all toposes $\mathcal{F},\mathcal{G}$ the category $Hom(\mathcal{F}\times\mathcal{E},\mathcal{G})$ is (naturally) equivalent as a category to $Hom(\mathcal{F},\mathcal{G}^\mathcal{E})$.

• By setting $\mathcal{F}=\mathcal{S}$ , the base topos, one sees from $Hom(\mathcal{S},\mathcal{G}^\mathcal{E})=Hom(\mathcal{S}\times\mathcal{E},\mathcal{G})=Hom(\mathcal{E},\mathcal{G})$ that $\mathcal{G}^\mathcal{E}$ is indeed a “mapping space” whose points are the geometric morphisms $\mathcal{E}\to\mathcal{G}$.

• The concept generalizes to higher topos theory (cf. Anel 2015, Anel-Lejay 2018, Lurie 2018).

Properties

Interestingly, in the category of locales exponentiability of a locale $X$ hinges on the existence of the single exponential $S^X$ where $S$ is the Sierpinski space: $Y^X$ exists for all $Y$ iff $S^X$ exists.

In $GrTop$ the object classifier $\mathcal{S}[\mathbb{O}]$ takes over the role of the Sierpinski space and we have the following

Proposition

A Grothendieck topos $\mathcal{E}$ is exponentiable iff the exponential $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ exists.

This result is due to Johnstone-Joyal (1982, p.282) and occurs as theorem 4.3.1 of Johnstone (2002, vol.1 p.433).

The following theorem pursues this analogy and generalizes a result of Martin Hyland on locales (1981).

Theorem

A Grothendieck topos $\mathcal{E}$ is an exponentiable object in the 2-category of Grothendieck toposes and geometric morphisms iff $\mathcal{E}$ is a continuous category.

This result is due to Johnstone-Joyal (1982) and occurs as theorem 4.4.5 of Johnstone (2002, p.748).

Exponentiability is a local property:

Proposition

1. If a Grothendieck topos $\mathcal{E}$ is exponentiable so is $\mathcal{E}/X$ for any object $X\in\mathcal{E}$.
2. If $\mathcal{E}/X$ is exponentiable and $X\to 1$ is an epimorphism, then $\mathcal{E}$ is exponentiable as well.

This occurs as lemma 4.2 in Johnstone-Joyal (1982, p.281).

Continuity also leaves a lattice-theoretic trace:

Proposition

If a Grothendieck topos $\mathcal{E}$ is exponentiable then the lattice of subobjects of any object $X\in\mathcal{E}$ is continuous.

This occurs as lemma 5.1 in Johnstone-Joyal (1982, p.287).

Examples

• Since locally finitely presentable categories are continuous and coherent toposes are locally finitely presentable (cf. Johnstone (2002, p.915)) it follows that coherent toposes are exponentiable. This can be viewed as an avatar of the fact that (locally) compact topological spaces behave well with respect to mapping spaces.

• By the same reasoning all functor categories $Set^{\mathcal{C}}$ for $\mathcal{C}$ a small category are exponentiable since they are locally finitely presentable. This includes in particular all presheaf toposes on small categories.

• An example of the latter is the Sierpinski topos $\mathcal{S}^2$. Here the exponential $\mathcal{F}^{\mathcal{S}^2}$ classifies the theory of $\mathbb{T}$-model homorphisms (with $\mathbb{T}$ the geometric theory classified by $\mathcal{F}$) i.e. the theory $\mathbb{T}^2$ such that $Mod_{\mathbb{T}^2}(\mathcal{E})=Mod_\mathbb{T}(\mathcal{E})^2$.

• The Malitz-Gregory topos? is locally finitely presentable but has not enough points (cf. Marquès 2024) whence it shows that there are exponentiable toposes failing to have enough points as well.

Remarks on duality

It is worthwhile to muse a bit about $\mathcal{S}[\mathbb{O}]^\mathcal{E}$:

First of at all, it is injective since $\mathcal{S}[\mathbb{O}]$ is injective and $(_-)^\mathcal{E}$ preserves inclusions.

This and the various universal properties of the toposes involved imply that an exponentiable topos

$$\mathcal{E}\cong Hom(\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{S},\mathcal{S}[\mathbb{O}]^\mathcal{E})$$

is (up to equivalence) the category of points of an injective topos.

Now consider an arbitrary topos $\mathcal{E}$ classifying a geometric theory $\mathbb{T}$.

Then from the functorial perspective on logic the 2-functor $\mathbb{T}^*:GrTop^{op}\to CAT$ assigning to $\mathcal{F}$ the category of models $\mathbb{T}^*(\mathcal{F}):=\mathcal{F}\times\mathcal{E}=\mathcal{F}\times\mathcal{S}[\mathbb{T}]$ is called the dual theory of $\mathbb{T}$.

It need not be geometric i.e. have a classifying topos but when it is, $\mathbb{T}$ being called dualizable in that case, the following

$$\mathbb{T}^*(\mathcal{F})=\mathcal{F}\times\mathcal{E}\cong Hom(\mathcal{F}\times\mathcal{E},\mathcal{S}[\mathbb{O}])\cong Hom(\mathcal{F},\mathcal{S}[\mathbb{O}]^\mathcal{E})$$

shows that $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ has precisely the properties required for $\mathcal{S}[\mathbb{T}^*]$.

In other words, $\mathbb{T}^*$ is geometric (and classified by $\mathcal{S}[\mathbb{O}]^\mathcal{E}$) precisely iff $\mathcal{E}=\mathcal{S}[\mathbb{T}]$ is exponentiable!

Compare also the remarks of Anel (2015) on the $\infty$-case.

Examples

• Let $0$ be the initial topos that classifies the inconsistent theory $\mathbb{T}^\emptyset_1$ over the empty signature. Then from $\mathcal{S}[\mathbb{O}]^0=\mathcal{S}$ follows that the dual of the inconsistent theory is $\mathbb{T}_1^{\emptyset\ast}=\mathbb{T}_{\emptyset}^{\emptyset}$ with $\mathbb{T}_{\emptyset}^{\emptyset}$ the empty theory (over the empty signature).

• From $\mathcal{S}[\mathbb{O}]^\mathcal{S}=\mathcal{S}[\mathbb{O}]$ follows in turn that the double dual of the inconsistent theory, or, in other words, the dual of the empty theory, is $\mathbb{T}_1^{\emptyset\ast\ast}=\mathbb{T}^{\emptyset\ast}_\emptyset=\mathbb{O}$ with $\mathbb{O}$ the theory of objects. (More generally, the dual theory of any dualizable theory is itself dualizable.)

• Let $\mathcal{S}^2$ be the Sierpinski topos that classifies subterminal objects or the theory of completely prime filters of the frame of opens of the Sierpinski space. Since $\mathcal{E}\times \mathcal{S}^2\cong \mathcal{E}^2$ it follows that

  $$\mathcal{E}^2=Hom(\mathcal{E}^2,\mathcal{S}[\mathbb{O}])=Hom(\mathcal{E}\times\mathcal{S}^2,\mathcal{S}[\mathbb{O}])= Hom(\mathcal{E},\mathcal{S}[\mathbb{O}]^{\mathcal{S}^2})=Hom(\mathcal{E},\mathcal{S}[\mathbb{O}^2])\quad ,$$

  i.e. the dual theory is the theory of morphisms $\mathbb{O}^2$.

The theory of sheaves on a coherent site

Recall, that given a small site $(\mathcal{C},J)$ there is a geometric theory $\mathbb{F}_J$ called the theory of J-continuous flat functors that (albeit uneconomically) axiomaticizes the theory classified by $Sh(\mathcal{C},J)$ using the objects and morphisms in $\mathcal{C}$ for its signature. When $Sh(\mathcal{C},J)$ is exponentiable this theory is dualizable, and, in the case the site $(\mathcal{C},J_c)$ is coherent i. e. $\mathcal{C}$ has finite limits and $J_c$ is generated by finite covering families, the following theory $\mathbb{S}$ called the theory of sheaves on $(\mathcal{C},J_{c})$ axiomatizes the dual theory $\mathbb{F}_{J_{c}}^\ast$ (in an equally uneconomic way):

The signature of $\mathbb{S}$ has a type symbol $A'$ for every object $A\in\mathcal{C}$ and a function symbol $f':B'\to A'$ for every morphism $f:B\to A$ in $\mathcal{C}^{op}$. The axioms are given by the following schemata

• $\top\vdash_x i'(x)=x$ for all identity morphisms $i$.

• $\top\vdash_x f'(x)=h'(g'(x))$ for all composable triples $f=g\circ h$ in $\mathcal{C}$.

• $\bigwedge_{i=1}^{n} (\alpha_i'(x)=a_i'(x'))\vdash_{x,x'} x=x'$ for each generating finite $J_{c}$-covering family $(Y_i\overset{\alpha_i}{\to} X|i=1,\dots, n)$.

• $\bigwedge_{i,j}(\beta_{ij}'(y_i)=\gamma_{ij}'(y_j))\vdash_{y_i,y_j}\exists x\big ( \bigwedge_{i=1}^{n}\alpha_i'(x)=y_i\big )$ for each generating finite $J_{c}$-covering family $(Y_i\overset{\alpha_i}{\to} X|i=1,\dots, n)$ and pullback

  $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \array{ Z_{ij} &\overset{\beta_{ij}}{\longrightarrow}& Y_i \\ {}^{\mathllap{\gamma_{ij}}} \downarrow & & \downarrow^{\mathrlap{\alpha_i}} \\ Y_j &\underset{\alpha_j}{\longrightarrow}& X }$ $\qquad \qquad.$

The models of $\mathbb{S}$ in a Grothendieck topos $\mathcal{F}$ are “sheaves of $\mathcal{F}$-objects” on $(\mathcal{C},J_{c})$ i.e. functors $\mathcal{C}^{op}\to\mathcal{F}$ satisfying the diagrammatic form of the sheaf axioms but

$$Mod_\mathbb{S}(\mathcal{F})=\mathcal{F}\times Sh(\mathcal{C},J_{c})=Hom(\mathcal{F}\times Sh(\mathcal{C},J_{c}), \mathcal{S}[\mathbb{O}])=Hom(\mathcal{F},\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})})$$

i.e. $\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})}$ is the classifying topos for $\mathbb{S}$. (For the details cf. Johnstone 1977, p.248f).

In particular, since $\mathbb{S}$ is a coherent theory this implies that $\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})}$ is also a coherent topos. From

$$Hom(\mathcal{S},\mathcal{S}[\mathbb{O}]^{Sh(\mathcal{C},J_{c})})=Mod_\mathbb{S}(\mathcal{S})=\mathcal{S}\times {Sh(\mathcal{C},J_{c})}=Sh(\mathcal{C},J_c)$$

it follows then that every coherent topos $Sh(\mathcal{C},J_c)$ is the category of points of a coherent topos.

Ramifications

The notion of a tiny object in a cartesian closed category suggests the following

Definition

An exponentiable Grothendieck topos $\mathcal{E}$ is called tiny (or infinitesimal) if the 2-functor $(_{-})^\mathcal{E}$ has a right 2-adjoint $(_{-})_\mathcal{E}$.

Related entries

• exponential object

• continuous category

• reflexive object

• injective topos

• ind-object

• classifying topos for the theory of objects

• cartesian closed 2-category

• convenient category of spaces

• Verdier duality

References

• Mathieu Anel, Toposes are commutative rings , talk at Topos à l’IHES, IHES Paris, Nov. 25-27, 2015 (video recording, at min 52:47 ff.).

• Mathieu Anel, Damien Lejay, Exponentiable Higher Toposes , arXiv:1802.10425 (2018). (abstract)

• Andreas Blass, The interaction of category theory and set theory , Cont. Math. 30 (1984) pp.5-29. (draft)

• Martin Hyland, Function spaces in the category of locales , Springer LNM 871 (1981) pp.264-281.

• Peter Johnstone, Topos Theory , Academic Press New York 1977. (Also available as Dover reprint Mineola 2014; pp.209-210, 248-249)

• Peter Johnstone, Sketches of an Elephant vols. 1&2 , CUP 2002. (sections B4.3 pp.432-438, C4.4 pp.745-754)

• Peter Johnstone, André Joyal, Continuous categories and exponentiable toposes , JPAA 25 (1982) pp.255-296.

• Jacob Lurie, Spectral Algebraic Geometry , ms. Harvard University 2018. (section 21.1.6)

• Jérémie Marquès?, Atomic Toposes with Co-Well-Founded Categories of Atoms , arXiv:2406.14346 (2024). (abstract)

• Susan Niefield, Exponentiable Morphisms: posets, spaces, locales and Grothendieck toposes , TAC 8 (2001) pp.16-32. (abstract)

• Myles Tierney, Forcing Topologies and Classifying Toposes , pp.211-219 in Heller, Tierney (eds.), Algebra, Topology and Category Theory , Academic Press New York 1976. (p.211, 218)